Confidence Regions in Wasserstein Distributionally Robust Estimation

Blanchet, José; Murthy, Karthyek; Si, Nian

doi:10.48550/arxiv.1906.01614

Cited by 6 publications

(6 citation statements)

References 31 publications

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“…, where the second inequality follows from Assumption 5 and [23, Lemma 2] (see also [24]). The desired result then readily follows from inequalities (15) and (17).…”

Section: A2 Proof Of Theoremmentioning

confidence: 84%

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Residuals-based distributionally robust optimization with covariate information

Kannan¹,

Bayraksan²,

Luedtke³

2020

Preprint

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We consider data-driven approaches that integrate a machine learning prediction model within distributionally robust optimization (DRO) given limited joint observations of uncertain parameters and covariates. Our framework is flexible in the sense that it can accommodate a variety of learning setups and DRO ambiguity sets. We investigate the asymptotic and finite sample properties of solutions obtained using Wasserstein, sample robust optimization, and phi-divergence-based ambiguity sets within our DRO formulations, and explore cross-validation approaches for sizing these ambiguity sets. Through numerical experiments, we validate our theoretical results, study the effectiveness of our approaches for sizing ambiguity sets, and illustrate the benefits of our DRO formulations in the limited data regime even when the prediction model is misspecified.

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“…, where the second inequality follows from Assumption 5 and [23, Lemma 2] (see also [24]). The desired result then readily follows from inequalities (15) and (17).…”

Section: A2 Proof Of Theoremmentioning

confidence: 84%

“…) results in estimators with a finite sample certificate-type guarantee (cf. [15]). This larger choice of the radius ζ n also yields estimators with the conventional O p (n −1/2 ) rate of convergence when we use parametric regression methods to estimate the function f * .…”

Section: Wasserstein-based Ambiguity Setsmentioning

confidence: 99%

Residuals-based distributionally robust optimization with covariate information

Kannan¹,

Bayraksan²,

Luedtke³

2020

Preprint

View full text Add to dashboard Cite

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“…where C 1 (θ) is the standard deviation of the loss function, Var 0 (l(θ, ξ)), multiplied by a constant that depends on φ. Similarly, if D is the Wasserstein distance and η is of order 1/n, (4) holds with C 1 (θ) being the gradient norm or the Lipschitz norm (Blanchet et al 2019a, Gao et al 2017, Blanchet et al 2019b). Furthermore, Staib and Jegelka (2019), which is perhaps closest to our work, studies MMD as the DRO statistical distance and derives a high-probability bound for Z(θ) similar to the RHS of (4), with C 1 (θ) being the reproducing kernel Hilbert space (RKHS) norm of l(θ, •).…”

Section: Variability Regularizationmentioning

confidence: 99%

Generalization Bounds with Minimal Dependency on Hypothesis Class via Distributionally Robust Optimization

Zeng¹,

Lam²

2021

Preprint

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Established approaches to obtain generalization bounds in data-driven optimization and machine learning mostly build on solutions from empirical risk minimization (ERM), which depend crucially on the functional complexity of the hypothesis class. In this paper, we present an alternate route to obtain these bounds on the solution from distributionally robust optimization (DRO), a recent data-driven optimization framework based on worst-case analysis and the notion of ambiguity set to capture statistical uncertainty. In contrast to the hypothesis class complexity in ERM, our DRO bounds depend on the ambiguity set geometry and its compatibility with the true loss function. Notably, when using maximum mean discrepancy as a DRO distance metric, our analysis implies, to the best of our knowledge, the first generalization bound in the literature that depends solely on the true loss function, entirely free of any complexity measures or bounds on the hypothesis class.

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“…As an effective way for the decision-making under uncertainty without fully knowing the probability distribution, distributionally robust optimization has attracted much attention [8,10,12,16,21,22,41,43,46]. Interested readers are referred to [34] for a comprehensive review.…”

Section: Relevant Literaturementioning

confidence: 99%

“…(a) If we focus on convergence of the optimal objective value instead of probability distribution, then the Wasserstein radius should be chosen in the order of O(N − 1 2 ) rather than O(N − 1 n ) shown in [14,15]. (b) The asymptotic rate O(N − 1 2 ) has been observed in [8] from a different angle. However, our rate is non-asymptotic.…”

Section: Proofmentioning

confidence: 99%

Distributionally Robust Bottleneck Combinatorial Problems: Uncertainty Quantification and Robust Decision Making

Xie

Zhang

Ahmed

2020

Preprint

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In a bottleneck combinatorial problem, the objective is to find a subset to minimize its highest cost of elements from the combinatorial solution space. This paper studies data-driven distributionally robust bottleneck combinatorial problems (DRBCP) with stochastic costs, where the probability distribution of the cost vector is contained in a ball of distributions centered at the empirical distribution specified by the Wasserstein distance. We study two distinct versions of DRBCP from different applications: (i) Motivated by the multi-hop wireless network application, we first study the uncertainty quantification of DRBCP (denoted by DRBCP-U), where decision-makers would like to have an accurate estimation of the worst-case value of DRBCP. The difficulty of DRBCP-U is to handle its max-min-max form. Fortunately, similar to the strong duality of linear programming, the alternative forms of the bottleneck combinatorial problems from their blockers allow us to derive equivalent deterministic reformulations, which can be computed via mixed-integer programs. In addition, by drawing the connection between DRBCP-U and its sampling average approximation counterpart under empirical distribution, we show that the Wasserstein radius can be chosen in the order of negative square root of sample size, improving the existing known results; and (ii) Next, motivated by the ride-sharing application, decision-makers choose the best service-andpassenger matching that minimizes the unfairness. This gives rise to the decision-making DRBCP (denoted by DRBCP-D). For DRBCP-D, we show that its optimal solution is also optimal to its sampling average approximation counterpart, and the Wasserstein radius can be chosen in a similar order as DRBCP-U. When the sample size is small, we propose to use the optimal value of DRBCP-D to construct an indifferent solution space and propose an alternative decision-robust model, which finds the best indifferent solution to minimize the empirical variance. We further show that the decision robust model can be recast as a mixedinteger program. Finally, we extend the proposed models and solution approaches to the distributionally robust Γ −sum bottleneck combinatorial problem (DRΓ BCP), where decision-makers are interested in minimizing the worst-case sum of Γ highest costs of elements.

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Confidence Regions in Wasserstein Distributionally Robust Estimation

Cited by 6 publications

References 31 publications

Residuals-based distributionally robust optimization with covariate information

Residuals-based distributionally robust optimization with covariate information

Generalization Bounds with Minimal Dependency on Hypothesis Class via Distributionally Robust Optimization

Distributionally Robust Bottleneck Combinatorial Problems: Uncertainty Quantification and Robust Decision Making

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